#### 12th Standard Maths English Medium Theory of Equations Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. A zero of x3 + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

2. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

(a)

mn

(b)

m+n

(c)

mn

(d)

nm

3. The number of real numbers in [0,2π] satisfying sin4x-2sin2x+1 is

(a)

2

(b)

4

(c)

1

(d)

4. If x3+12x2+10ax+1999 definitely has a positive zero, if and only if

(a)

a≥0

(b)

a>0

(c)

a<0

(d)

a≤0

5. The number of positive zeros of the polynomial $\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }$(-1)rxr is

(a)

0

(b)

n

(c)

< n

(d)

r

6. If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax2+bx+c=0 then the other root is

(a)

-p+√q

(b)

p-iq

(c)

p-√q

(d)

-p-√q

7. Let a > 0, b > 0, c >0. h n both th root of th quatlon ax2+b+C= 0 are

(a)

real and negative

(b)

real and positive

(c)

rational numb rs

(d)

none

8. lf the root of the equation x3 +bx2+cx-1=0 form an lncreasing G.P, then

(a)

one of the roots is 2

(b)

one of the rots is 1

(c)

one of the rots is -1

(d)

one of the rots is -2

9. For real x, the equation $\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| }$ has

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

10. If the equation ax2+ bx+c=0(a>0) has two roots ∝ and β such that ∝<- 2 and β > 2, then

(a)

b2-4ac=0

(b)

b2 - 4ac <0

(c)

b2 - 4ac >0

(d)

b2 - 4ac≥0

11. If (2+√3)x2-2x+1+(2-√3)x2-2x-1=$\frac { 2 }{ 2-\sqrt { 3 } }$ then x=

(a)

0,2

(b)

0,1

(c)

0,3

(d)

0, √3

12. If ∝, β,૪ are the roots of 9x3-7x+6=0, then ∝ β ૪ is __________

(a)

$\frac{-7}{9}$

(b)

$\frac{7}{9}$

(c)

0

(d)

$\frac{-2}{3}$

13. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

(a)

0

(b)

1

(c)

4

(d)

3

14. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

15. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

1. 2 Marks

10 x 2 = 20
16. If α, β, γ  and $\delta$ are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + $\delta$ and αβ૪$\delta$.

17. Show that the equation 2x2−6x+7=0 cannot be satisfied by any real values of x.

18. If x2+2(k+2)x+9k=0 has equal roots, find k.

19. Solve: (2x-1)(x+3)(x-2)(2x+3)+20=0

20. Determine the number of positive and negative roots of the equation x9-5x4-14x7=0.

21. Construct a cubic equation with roots 1,1, and −2

22. Construct a cubic equation with roots 2,−2, and 4.

23. Discuss the nature of the roots of the following polynomials:
x5-19x4+2x3+5x2+11

24. Find th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

25. Find x If $x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } }$

1. 3 Marks

10 x 3 = 30
26. If α and β are the roots of the quadratic equation 2x2−7x+13 = 0 , construct a quadratic equation whose roots are α2 and β2.

27. If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

28. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are, 2α, 2β, 2γ

29. Find the sum of squares of roots of the equation 2x4-8x+6x2-3=0.

30. Find the condition that the roots of x3+ax2+bx+c = 0 are in the ratio p:q:r.

31. If p is real, discuss the nature of the roots of the equation 4x2+4px+p+2=0 in terms of p.

32. If α, β, and γ are the roots of the polynomial equation ax3+bx2+cx+d=0 , find the value of $\Sigma \frac { \alpha }{ \beta \gamma }$ in terms of the coefficients.

33. Solve the cubic equations:
2x3-9x2+10x=3

34. Solve the following equations,
12x+8x=29x2-4

35. Solve the equations
x4+3x3-3x-1=0

1. 5 Marks

7 x 5 = 35
36. Form the equation whose roots are the squares of the roots of the cubic equation x3+ax2+bx+c = 0.

37. Solve the equation x3−9x2+14x+24=0 if it is given that two of its roots are in the ratio 3:2.

38. If 2+i and 3-$\sqrt{2}$ are roots of the equation x6-13x5+62x4-126x3+65x2+127x-140=0, find all roots.

39. Solve:
(x-5)(x-7)(x+6)(x+4)=504

40. If a, b, c, d and p are distinct non-zero real numbers such that (a2+b2+c2) p2-2 (ab+bc+cd) p+(b2+c2+d2)≤ 0 the n. Prove that a,b,c,d are in G.P and ad=bc

41. If c ≠ 0 and $\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c }$ has two equal roots, then find p.

42. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.